Easy Cases \(\psi _0,\psi _r\). To begin with, the cases \(i = 0,r\)  can be characterized almost immediately. When \(i = 0\)  i.e., \(\psi _i\)  is the predicate which accepts if and only if the string is all zeros. Then the all zeros labeling would satisfy all constraints, and hence this trivially yields a 1-approximation algorithm for \(\psi _0\) . On the other hand when \(i = r\) , then \(\psi _i\)  corresponds to the \({\sf AND}\)  predicate, and in particular expresses the Densest-\(k\) -SubHypergraph (\({\sf D}k{\sf SH}\) ) problem. Since for \(r = 2\) , there exists a \(\mu \) -approximation algorithm [1]} and SSEH impiles that \({\sf D}k{\sf S}\)  is constant factor inapproximable [2]}, it follows that \(\lim _{\mu \rightarrow 0} \alpha _{(\mu )}(\psi _r) = 0\) .
